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DIFFERENTIAL CALCULUS
Study Notes 6

“FUNCTIONS”

Prepared By: Tutor Win

FUNCTIONS

08/02/2021

In the last StudyNote 5, we have learned the different sections of the
cone such as the parabola, ellipse, and hyperbola
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We have studied the
properties underlying these concepts to help us understand the
fundamentals of Calculus
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This basic concept of functions helps us further develop of critical and
analytical skills in solving advanced math problems in the future topics
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The
dependent variable literally depends on the dependent variable
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The value of the
output y could be calculated because of the value of the input x
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The notation of a function is f(x), read as “the function of x
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The domain here is the set of all
real numbers
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Let’s say we will input x = 1
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The coordinates of the point are (1,5)
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Then, plot these points to the
rectangular coordinate system to graph the function
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Find f(-3) of the function f(x) = x2 – 6x – 9
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f(x) = x2 – 6x – 9
Step 2
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f(-3) = 9 + 18 – 9
Step 4
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Find f(2) of the function f(x) = 5x2 +3x – 1
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f(x) = 5x2 +3x – 1
Step 2
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Step 4
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EXAMPLE 3
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Step 2
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To find the domain or inputs, take note that these are
values that will not make y undefined
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Answer:
x is the set of all real numbers
EXAMPLE 4
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Step 2
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Find the possible situation that will make y undefined
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That is, 5x2 ≥ 0 in order for y to be defined
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5-x2 ≥ 0
x2 ≥ 5
Answer:

EXAMPLE 5
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Step 2
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Find the possible situation that will make y undefined
relating to square root
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That is, x2 – 25 ≥ 0 in order for y to be
defined
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x2 – 25 ≥ 0
x2 ≥ 25
Answer:
x≤5
and
x≥5

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FUNCTIONS

EXAMPLE 6
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Step 2
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Find the possible situation that will make y undefined
relating to rational function
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That is, x – 8 ≠ 0 in
order for the function to be defined
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x–8=0
x–8+8=0+8
x=8
The value of x should not be equal to 8 in order for the y
to be defined
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If we will
input x to the function, we will get the output y
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EXAMPLE 7
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x
-3
y
Step 2
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Assign x-values including negative and positive integers
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Just substitute each
one to the function then solve for y
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08/02/2021

Plot these points on the rectangular coordinate system
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Other method is to get the slope
and intercept, then plot these on the rectangular coordinate system
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EXAMPLE 8
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x
y

-3
Step 2
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Assign x-values including negative and positive integers
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Just substitute each
one to the function then solve for y
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4|Page

FUNCTIONS

08/02/2021

Note that this is a parabola
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5|Page

FUNCTIONS

08/02/2021

PRACTICE TEST 6
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Find what is asked
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f(x) =
Find a
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f(3), c
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f(2x), e
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f(x) = x4 – 5x3 – 2x2 + x + 4
Find a
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f(3), c
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f(2x), e
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Find the domain of the following functions
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y = √2 − 𝑥
4
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y =
6
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y =

C
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8
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y = x2 + 5x
10
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SOLUTIONS

A
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a
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f(3) =

𝟔

( )

f(3) =
f(3) =

𝟗
𝟏𝟒

c
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f(2x) =
f(2x) =

𝟑
𝟏𝟒

( )
𝟐𝒙 𝟔
𝟒𝒙𝟐 𝟓

e
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a
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f(3) = (3)4 – 5(3)3 – 2(3)2 + (3) + 4
f(3) = 81 – 5(27) – 2(9) + 3 + 4
f(3) = 81 – 135 – 18 + 3 + 4
f(3) = -65
c
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f(2x) = (2x)4 – 5(2x)3 – 2(2x)2 + (2x) + 4
f(2x) = 16x4 – 5(8x3)– 2(4x2) + 2x + 4
f(2x) = 16x4 – 40x3 – 8x2+ 2x + 4
e
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3
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2-x2 ≥ 0
x2 ≥ 2
Answer: -√𝟐 ≤ x ≤ √𝟐
x2 – 49 ≥ 0
x2 ≥ 49
Answer:
x≤7

and

x≥7

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FUNCTIONS

08/02/2021

5
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x2 – 4 ≠ 0
(x-2)(x+2) = 0
x–2=0
x+2=0
x=2
x = -2
Answer: all real numbers except 2 and -2

7
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Answer: all real numbers

C
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x
y

-3
-2
-13
-9
Answer:

-1
-5

0
-1

1
3

2
7

3
11

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FUNCTIONS

08/02/2021

9
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x
y

-3
-20
Answer:

10 | P a g e



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